%! program = pdflatex
\section{Notes on Intro section}

The main ideas of the paper are three-fold:
\begin{itemize}
\item To present a general framework under which active sampling of
Lagrangian sensors and estimation under uncertainties for obtaining
macroscopic quantities can be done. (See Figure 2.)

\item To investigate active sensor selection methods and to compare
performance of 2-3 estimation algorithms on first-order models: e.g., robust estimation algorithms, 
particle filters, hybrid system based estimation algorithms, EnKF, etc.
This is supposed to the main technical part of the paper.

\item To present experimental results using NGSIM data and century data.

\end{itemize}

Traffic flow on a road network is a spatio-temporal phenomenon. Accurate and efficient determination of macroscopic traffic flow characteristics is necessary for traffic operations and management. The quantities of interest are\footnote{These definitions are documented in the Highway Capacity Manual. More general definitions of traffic flow characteristics exist in literature. For example, Edie's definitions.}: 
\begin{itemize}
\item Flow, $Q(x,t)$, which is direct measure of throughput defined as number of vehicles passing a point on a highway during unit duration of time, 
\item Density, $K(x,t)$,  which is an indicator of traffic conditions (congested vs. uncongested) defined as number of vehicles observed on a segment over unit length of space,
\item Space mean speed (SMS), $V(x,t)$, which is useful for computing travel times defined as the harmonic mean of speeds over a given segment of road or an average speed based on the average travel time of vehicles to traverse a given segment of road.
\end{itemize}

In contrast to the macroscopic level, the microscopic flow characteristics include the position~$x_{i}(t)$, velocity~$v_{i}(t)=\dot x(t)$, acceleration~$a_{i}(t)=\ddot x_{i}(t)$ and spacing $s_{i}(t)=x_{i+1}(t)-x_{i}(t)$. One of the problems of interest to us is to determine macroscopic characteristics from the results of microscopic analysis. 

There are a wide variety of sensors that are available for determining traffic flow characteristics. Sensing has three dimensions: space, time, and the number of vehicles. If we are able to measure the space-time trajectories of all the vehicles in the highway (as is the case with the NGSIM data), then we can exactly determine all the traffic flow characteristics. However, the current sensing techniques only measure two of the above three dimensions, thus requiring some sort of approximation for determining the third dimension. This is where models of the flow phenomenon are useful. 

Among the available sensing techniques, we distinguish between point sensors such as loop detectors and video cameras, space sensors such as aerial photography and satellite data, and mobile sensors such as GPS equipped vehicles and automatic vehicle location techniques. 

%\begin{figure}
%\centering
%\includegraphics[width=4in]{{figures/data}
%%picclQX11
%\caption{Sensing techniques}
%\label{fig:}
%\end{figure}

\begin{figure}[htbp]
\centerline{
\includegraphics[width=4in]{figures/data}}
\caption{Sensing techniques.}
\label{sensors}
\end{figure}

\begin{itemize}
\item \emph{Point sensors} are fixed in location along a roadway and measure vehicles passing through this location throughout the time for which they are active. Therefore, point sensor ignore the space dimension.

\item \emph{Space sensors} can take snapshots of traffic at a given instant of time and repeat such snapshots are multiple time instants. However, such snapshots can be taken only for limited time instants and this, space sensors ignore the time dimension.

\item \emph{Mobile sensors} move with the traffic flow in space-time domain and can collect the time stamped position (and hence the speed) of the vehicle. However, due to a limited penetration rate of mobile sensors and due to availability and privacy constraints, mobile sensor data is only restricted to small sample of vehicles. Thus, mobile sensors ignore the number-of-vehicles dimension.
\end{itemize}

It is clear that all the current traffic sensing methods only acquire samples of the traffic flow phenomenon at discrete points in space and/or time. It is not possible to completely gather all information about traffic flow over the entire freeway for all times. On the contrary, all the data that is gathered might not be of direct use to address user specific queries. Furthermore, the acquisition of measurements is not uniform due to the fact that the sensors are not uniformly deployed over space and can intermittently fail over time. For real time applications, the traffic flow itself is a highly complicated phenomenon and considerable effort has been placed to reliably model the evolution of traffic on a freeway. Due to these reasons, crude averaging and interpolation methods, although simple and efficient for real-time querying applications, are most likely a bad idea for accurately determining the variables related to traffic flow. We next propose a general framework for real-time traffic state estimation on freeways using available sensor data. 

%\section{Framework for sensing and estimation}

%\begin{figure}
%\centering
%\includegraphics[width=5in]{figures/framework}
%%picclQX11
%\caption{Framework for sensing and estimation}
%\label{fig:framework}
%\end{figure}
\section{Framework for Sensing and Estimation}\label{sec_framework}

We address the problem of real-time traffic state estimation on freeways into to sub-problems:

\begin{itemize}
\item[(A)] Given a set of user queries, a traffic flow model, and available data in the form of localized averages of macroscopic flow variables in small space-time sections as well as estimates of demands on a freeway, how to optimally assimilate or filter the data in real-time to best address the user queries. This requires development of estimation algorithms. 
\item[(B)] Given the constraints imposed by sensing techniques, for example, availability and privacy constraints, and the "demand of measurements", how to derive an optimal data acquisition or sensor selection plan such that raw data measurements can be used to arrive at localized averages of flow variables to be used by the estimation algorithms. This requires development of sensor selection algorithms. 
\end{itemize}


\begin{figure}[htbp]
\centerline{
\includegraphics[width=4in]{figures/framework}}
\caption{Framework for sensing and estimation.}
\label{framework}
\end{figure}



%
%\subsection{Comparison with PeMS and other existing traffic sensing services}
%
%
%\section{(Active) Sensing}
%
%Lagrangian/particle flow equation,
%Arriving at macroscopic estimate of speed from microscopic data,
%
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%\subsection{Sampling}
%
%
%\subsection{Active sensor selection}
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%\subsubsection{Availability constraints}
%- 
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%\subsubsection{Privacy constraints}
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%- 
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%\section{Dynamic Model based estimation}
%
%
%\subsection{First-order flow model: CTM/LWR}
%
%\emph{Note}: Some of this section is taken from Kurzhanskiy's dissertation. 
%
%
%
%\subsection{Modeling uncertainties in demand and model parameters}  
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%\subsection{Estimation under set-valued uncertainty model}
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%\subsection{Piecewise affine representation}
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%\subsection{Estimation under probabilistic uncertainty model}
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%\subsection{Comparison with general Bayesian filters}
%
%- Comparison with EnKF,
%- Comparison with Particle filtering
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%
%\section{Analysis using NGSIM and Century data}
%
%- Experimental bounds on minimum penetration rate and characterization of worst-case sensing policy to achieve acceptable confident on estimates. 
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%\section{New challenges and future work}
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%\end{document}